Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
169 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 169 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #2206 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×D8 | C2 | semidirect |
| 2. #2240 | C2 | semidirect |
| 4. #1668 | C2 | semidirect |
| 5. #1364 | C2 | semidirect |
| 6. Q8⋊C16 | C2 | semidirect |
| 7. #1331 | C2 | semidirect |
| 69. D8 | SD16 | semidirect |
| 92. Q8 | D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D8 | 2 |
| #2240 | 2 |
| #2206 | 2 |
| #1668 | 2 |
| #1364 | 2 |
| Q8⋊C16 | 2 |
| #1331 | 2 |